Alexeev Unified Non-Local Theory of Transport Processes

Generalized Boltzmann Physical Kinetics
2. Auflage 2015
ISBN: 978-0-444-63487-0
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: 6 - ePub Watermark

E-Book, Englisch, 644 Seiten

Unified Non-Local Theory of Transport Processess, 2nd Edition provides a new theory of transport processes in gases, plasmas and liquids. It is shown that the well-known Boltzmann equation, which is the basis of the classical kinetic theory, is incorrect in the definite sense. Additional terms need to be added leading to a dramatic change in transport theory. The result is a strict theory of turbulence and the possibility to calculate turbulent flows from the first principles of physics. - Fully revised and expanded edition, providing applications in quantum non-local hydrodynamics, quantum solitons in solid matter, and plasmas - Uses generalized Boltzmann kinetic theory as an highly effective tool for solving many physical problems beyond classical physics - Addresses dark matter and energy - Presents non-local physics in many related problems of hydrodynamics, gravity, black holes, nonlinear optics, and applied mathematics

Professor Boris V. Alexeev is Head of the Centre of the Theoretical Foundations of Nanotechnology and Head of the Physics Department at the Moscow Lomonosov University of Fine Chemical Technologies, Moscow, Russia. In the 1990s he was Visiting Professor at the University of Alabama, Huntsville, AL, USA, and Visiting Professor at the University of Provence, Marseille, France. Professor Alexeev has published over 290 articles in international scientific journals and 22 books. He has received several honors and awards, and is member of six societies.

Autoren/Hrsg.

Alexeev, Boris V.

Inhaltsverzeichnis

1;Front Cover;1
2;Unified Non-Local Theory of Transport Processes: Generalized Boltzmann Physical Kinetics;4
3;Copyright;5
4;Contents;6
5;Preface;10
6;Historical Introduction and the Problem Formulation;12
7;Chapter 1: Generalized Boltzmann Equation;28
7.1;1.1. Mathematical Introduction-Method of Many Scales;28
7.2;1.2. Hierarchy of Bogolubov Kinetic Equations;38
7.3;1.3. Derivation of the Generalized Boltzmann Equation;42
7.4;1.4. Generalized Boltzmann H-Theorem and the Problem of Irreversibility of Time;58
7.5;1.5. Generalized Boltzmann Equation and Iterative Construction of Higher-Order Equations in the Boltzmann Kinetic Theory;70
7.6;1.6. Generalized Boltzmann Equation and the Theory of Non-Local Kinetic Equations with Time Delay;73
8;Chapter 2: Theory of Generalized Hydrodynamic Equations;80
8.1;2.1. Transport of Molecular Characteristics;80
8.2;2.2. Hydrodynamic Enskog Equations;82
8.3;2.3. Transformations of the Generalized Boltzmann Equation;83
8.4;2.4. Generalized Continuity Equation;85
8.5;2.5. Generalized Momentum Equation for Component;87
8.6;2.6. Generalized Energy Equation for Component;90
8.7;2.7. Summary of the Generalized Enskog Equations and Derivation of the Generalized Hydrodynamic Euler Equations;94
9;Chapter 3: Quantum Non-Local Hydrodynamics;104
9.1;3.1. Generalized Hydrodynamic Equations and Quantum Mechanics;104
9.2;3.2. GHEs, Quantum Hydrodynamics. SE as the Consequence of GHE;109
9.3;3.3. SE and its Derivation from Liouville Equation;115
9.4;3.4. Direct Experimental Confirmations of the Non-Local Effects;116
10;Chapter 4: Application of Unified Non-Local Theory to the Calculation of the Electron and Proton Inner Structures;124
10.1;4.1. Generalized Quantum Hydrodynamic Equations;124
10.2;4.2. The Charge Internal Structure of Electron;127
10.3;4.3. The Derivation of the Angle Relaxation Equation;133
10.4;4.4. The Mathematical Modeling of the Charge Distribution in Electron and Proton;135
10.5;4.5. To the Theory of Proton and Electron as Ball-like Charged Objects;156
11;Chapter 5: Non-Local Quantum Hydrodynamics in the Theory of Plasmoids and the Atom Structure;158
11.1;5.1. The Stationary Single Spherical Plasmoid;158
11.2;5.2. Results of the Mathematical Modeling of the Rest Solitons;160
11.3;5.3. Nonstationary 1D Generalized Hydrodynamic Equations in the Self-Consistent Electrical Field. Quantization in the Gen ...;168
11.4;5.4. Moving Quantum Solitons in Self-Consistent Electric Field;171
11.5;5.5. Mathematical Modeling of Moving Solitons;173
11.6;5.6. Some Remarks Concerning CPT (Charge-Parity-Time) Principle;183
11.7;5.7. About Some Mysterious Events of the Last Hundred Years;189
11.7.1;5.7.1. Tunguska Event (TE);189
11.7.2;5.7.2. Gagarin and Seryogin Air Crash;191
11.7.3;5.7.3. Accident with Malaysia Airlines Flight MH370;193
12;Chapter 6: Quantum Solitons in Solid Matter;196
12.1;6.1. Quantum Oscillators in the Unified Non-local Theory;196
12.2;6.2. Application of Non-Local Quantum Hydrodynamics to the Description of the Charged Density Waves in the Graphene Cryst ...;204
12.3;6.3. Generalized Quantum Hydrodynamic Equations Describing the Soliton Movement in the Crystal Lattice;206
12.4;6.4. Results of the Mathematical Modeling Without the External Electric Field;216
12.5;6.5. Results of the Mathematical Modeling With the External Electric Field;240
12.6;6.6. Spin Effects in the Generalized Quantum Hydrodynamic Equations;259
12.7;6.7. To the Theory of the SC;266
13;Chapter 7: Generalized Boltzmann Physical Kinetics in Physics of Plasma;270
13.1;7.1. Extension of Generalized Boltzmann Physical Kinetics for the Transport Processes Description in Plasma;270
13.2;7.2. Dispersion Equations of Plasma in Generalized Boltzmann Theory;277
13.3;7.3. The Generalized Theory of Landau Damping;281
13.4;7.4. Evaluation of Landau Integral;283
13.5;7.5. Estimation of the Accuracy of Landau Approximation;291
13.6;7.6. Alternative Analytical Solutions of the Vlasov-Landau Dispersion Equation;295
13.7;7.7. The Generalized Theory of Landau Damping in Collisional Media;302
14;Chapter 8: Physics of a Weakly Ionized Gas;312
14.1;8.1. Charged Particles Relaxation in ``Maxwellian´´ Gas and the Hydrodynamic Aspects of the Theory;312
14.2;8.2. Distribution Function (DF) of the Charged Particles in the ``Lorentz´´ Gas;315
14.3;8.3. Charged Particles in Alternating Electric Field;326
14.4;8.4. Conductivity of a Weakly Ionized Gas in the Crossed Electric and Magnetic Fields;328
14.5;8.5. Investigation of the GBE for Electron Energy Distribution in a Constant Electric Field with due Regard for Inelastic ...;333
15;Chapter 9: Generalized Boltzmann Equation in the Theory of the Rarefied Gases and Liquids;342
15.1;9.1. Kinetic Coefficients in the Theory of the Generalized Kinetic Equations. Linearization of the Generalized Boltzmann ...;342
15.2;9.2. Approximate Modified Chapman-Enskog Method;348
15.3;9.3. Kinetic Coefficient Calculation with Taking into Account the Statistical Fluctuations;356
15.4;9.4. Sound Propagation Studied with the Generalized Equations of Fluid Dynamics;360
15.5;9.5. Shock Wave Structure Examined with the Generalized Equations of Fluid Dynamics;372
15.6;9.6. Boundary Conditions in the Theory of the Generalized Hydrodynamic Equations;374
15.7;9.7. To the Kinetic and Hydrodynamic Theory of Liquids;379
16;Chapter 10: Strict Theory of Turbulence and Some Applications of the Generalized Hydrodynamic Theory;390
16.1;10.1. About Principles of Classical Theory of Turbulent Flows;390
16.2;10.2. Theory of Turbulence and Generalized Euler Equations;391
16.3;10.3. Theory of Turbulence and the Generalized Enskog Equations;400
16.4;10.4. Unsteady Flow of a Compressible Gas in a Cavity;404
16.5;10.5. Application of the GHE: To the Investigation of Gas Flows in Channels with a Step;414
16.6;10.6. Vortex and Turbulent Flow of Viscous Gas in Channel with Flat Plate;422
17;Chapter 11: Astrophysical Applications;440
17.1;11.1. Solution of the Dark Matter Problem in the Frame of the Non-Local Physics;440
17.2;11.2. Plasma-Gravitational Analogy in the Generalized Theory of Landau Damping;441
17.3;11.3. Disk Galaxy Rotation and the Problem of Dark Matter;452
17.4;11.4. Hubble Expansion and the Problem of Dark Energy;465
17.5;11.5. Propagation of Plane Gravitational Waves in Vacuum with Cosmic Microwave Background;472
17.6;11.6. Application of the Non-Local Physics in the Theory of the Matter Movement in Black Hole;488
17.7;11.7. Self-similar Solutions of the Non-local Equations;496
17.7.1;11.7.1. Preliminary Remarks;496
17.7.2;11.7.2. Self-similar Solutions of the Non-Local Equations in the Astrophysical Applications;499
18;Chapter 12: The Generalized Relativistic Kinetic Hydrodynamic Theory;520
18.1;12.1. Hydrodynamic Form of the Dirac Quantum Relativistic Equation;520
18.2;12.2. Generalized Relativistic Kinetic Equation;524
18.3;12.3. Generalized Enskog Relativistic Hydrodynamic Equations;529
18.3.1;12.3.1. Derivation of the Continuity Equation;529
18.3.2;12.3.2. Derivation of the Motion Equation;531
18.3.3;12.3.3. Derivation of the Energy Equation;533
18.4;12.4. Generalized System of the Relativistic Hydrodynamics and Transfer to the Generalized Relativistic non-Local Euler H ...;535
18.5;12.5. Generalized non-Local Relativistic Euler Equations;540
18.6;12.6. The Limit Transfer to the non-Relativistic Generalized non-Local Euler Equations;543
18.6.1;12.6.1. Some Auxiliary Expressions;543
18.6.2;12.6.2. Non-Relativistic Generalized Euler Equations as Asymptotic of the Relativistic Equations;544
18.7;12.7. Expansion of the Flat Harmonic Waves of Small Amplitudes in Ultra-relativistic Media;547
18.8;Some remarks to the conclusion of the monograph;558
19;Appendix 1: Perturbation Method of the Equation Solution Related to T[f];560
20;Appendix 2: Using of Curvilinear Coordinates in the Generalized Hydrodynamic Theory;564
21;Appendix 3: Characteristic Scales in Plasma Physics;584
22;Appendix 4: Dispersion Relations in the Generalized Boltzmann Kinetic Theory Neglecting the Integral Collision Term;586
23;Appendix 5: Three-Diagonal Method of Gauss Elimination Techniques for the Differential Third- and Second-Order Equations;588
24;Appendix 6: Some Integral Calculations in the Generalized Navier-Stokes Approximation;594
25;Appendix 7: Derivation of Energy Equation for Invariant Ea=maVa22+ea;596
26;Appendix 8: To the Non-Local Theory of Cold Nuclear Fusion;602
27; Appendix 9: To the Non-Local Theory of Variable Stars;610
28;Appendix 10: To the Non-Local Theory of Levitation;620
29;References;628
30;Index;636

Leseproben

Chapter 1 Generalized Boltzmann Equation
Abstract
In what follows, we intend to construct the generalized Boltzmann physical kinetics using the different methods of the kinetic equation derivations from the Bogolubov hierarchy. Keywords Generalized Boltzmann equation Bogolubov hierarchy Method of many scales 1.1 Mathematical Introduction—Method of Many Scales
In the sequel asymptotic methods will be used, but first of all, we will look at the method of many scales. The method of many scales is so popular that Nayfeh in his book [68] written more than 40 years ago said that method of many scales (MMS) is discovered by different authors every half a year. As a result, there exist many different variants of MMS. As a minimum four variants of MMS are considered in [68]. We are interested only in the main ideas of MMS, which are used further in theory of kinetic equations. From this standpoint we demonstrate MMS possibilities using typical example of solution linear differential equation which also has the exact solution for comparing the results [68]. But in contrast with usual consideration, which can be found in literature, we intend to bring this example up to table and a graph. Therefore let us consider the linear differential equation x¨+?x?+x=0.   (1.1.1) We begin with the special case when d = 1 and is a small parameter. Equation (1.1.1) has the exact solution =ae-?t/2cost1-14?2+?,   (1.1.2) where a and ? are arbitrary constants of integrating. In typical case of small parameter d in front of senior derivative—in this case it would be ¨—the effects of boundary layer can be observed. Using the derivatives x?=-12?x-ae-?t/21-14?2sint1-14?2+?,x¨=-12?x?-x1-14?2+12a?e-?t/21-14?2sint1-14?2+?, for substitution in Eq. (1.1.1) we find the identical satisfaction of Eq. (1.1.1). We begin with a direct expansion in small , using series =x0+?x1+?2x2+?,   (1.1.3) and after differentiating x?=x?0+?x?1+?2x?2+?,x¨=x¨0+?x¨1+?2x¨2+?. Substitute series (1.1.3) into (1.1.1) and equalize coefficients in front of equal powers of , having ¨0+x0=0,   (1.1.4) ¨1+x1=-x?0,   (1.1.5) ¨2+x2=-x?1,   (1.1.6) ¨3+x3=-x?2,   (1.1.7) and so on. The general solution of homogeneous Eq. (1.1.4) has the form 0=acost+?.   (1.1.8) Substitute (1.1.8) in (1.1.5): ¨1+x1=asint+?.   (1.1.9) General solution (1.1.1) should contain only two arbitrary constants. In this case, both constants a and ? are contained in the main term of expansion defined by relation (1.1.8). Then we need find only particular solution of Eq. (1.1.9); which can be found as follows 1=-12atcost+?.   (1.1.10) Really, x?1=-12acost+?+12atsin(t+?),x¨1=12asint+?-x1+12asin(t+?). After substitution in (1.1.9), we find identity. Equation (1.1.6) can be rewritten as ¨2+x2=12acost+?-12atsint+?,   (1.1.11) and its solution 2=18at2cost+?+18atsint+?.   (1.1.12) Really, ?2=14atcost+?-18at2sint+?+18asint+?+18atcost+?, ¨2=12acost+?-58atsint+?-18at2cost+?. Substitution into left-hand side of Eq. (1.1.11), lead to result 2acost+?-58atsin(t+?)-18at2cos(t+?)+18at2cost+?+18atsin(t+?)=12acost+?-12atsin(t+?). Then we state the identical satisfaction of Eq. (1.1.11) by solution (1.1.12). In analogous way the solution of Eq. (1.1.7) is written as cubic polynomial in t. For the first three terms of Eq. (1.1.3) series the solution is =acost+?-12?atcos(t+?)+18?2at2cost+?+tsint+?+O?3.   (1.1.13) At our desire the variable t can be considered as dimensionless time. Suppose, of course, that we wish to have a solution for arbitrary time moments. But it is not possible in the developed procedure, because the series construction regards the successive terms of the series to be smaller than the forgoing terms; in other case, it is impossible to speak about series convergence. But for fixed , the time moment can be found when successive term of expansion is no smaller than the forgoing term. Figure 1.1 contains comparison of the exact solution (1.1.2) for concrete parameters of calculations a = 1, ? = 0,  = 0.2 with approximate solutions 0=costx1=cost-0.1tcostx2=cost-0.1tcost+0.005t2cost+tsintxex=e-0.1tcost0.99. As we could expect, the divergence of solution 2x and exact solution exx appears when t is of order 10; or, in the common case, if t ~ - 1. But as it follows from Fig. 1.1, the situation is much worse, because, for example, for t = 4p approximate solution 1x gives wrong sign in comparison with the exact solution exx. For the mathematical model of an oscillator with damping—which is reflecting by Eq. (1.1.1), it means that approximate solution 1x forecasts a deviation in the opposite direction for the mentioned oscillator. By the way, solution 1x is also worse in comparison with solution 0x, in this sense the minor approximation is better than senior ones. Figure 1.1 Comparison of solutions 0x, 1x, 2x, obtained by perturbation method with the many scales solution msx2 and exact solution exx for the case a= 1, f = 0,  = 0.2. This poses the question how to improve the situation remaining in the frame of asymptotic methods. To answer this question, let us consider the exact solution (1.1.2). Exponential and cosine terms containing in this solution, can be expand in the following series for small and fixed t: -?t/2=1-12?t+18?t2-164?t3+?   (1.1.14) cost1-14?2+?=cost1-18?2-1128?4-11024?6-?+?=cost+?-18t?2-1128t?4-11024t?6-??cost+?+18t?2+1128t?4+11024t?6+?sin(t+?)=cost+?+18t?2sin(t+?)+1128t?4sin(t+?)+?.   (1.1.15) Obviously, product of the first terms in expansions (1.1.14) and (1.1.15) gives 0x, and retaining of terms of O(3) lead to result ?a1-12?t+18?2t2cost+?+18t?2sint+??acost+?-12a?tcos(t+?)+18at?2sin(t+?)+18a?2t2cos(t+?). Then we state that the used construction of asymptotic solution is based in deed on the assumption that combination ?t is small. If it is not so (for t having the order - 1), then expansions (1.1.14) and (1.1.15) are wrong or need to take into account all terms of expansions. But asymptotic expansion can be organized by another way, using additional variables: 1=?t   (1.1.16) and 2=?2t.   (1.1.17) In this case -?t/2=e-T1/2,   (1.1.18) and expansion (1.1.15) is replaced by other...